Integrand size = 23, antiderivative size = 23 \[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{9} (1+x)^{3/2} \left (1-x+x^2\right )^{3/2} \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {927} \[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{9} (x+1)^{3/2} \left (x^2-x+1\right )^{3/2} \]
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Rule 927
Rubi steps \begin{align*} \text {integral}& = \frac {2}{9} (1+x)^{3/2} \left (1-x+x^2\right )^{3/2} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{9} (1+x)^{3/2} \left (1-x+x^2\right )^{3/2} \]
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Time = 0.58 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(\frac {2 \left (1+x \right )^{\frac {3}{2}} \left (x^{2}-x +1\right )^{\frac {3}{2}}}{9}\) | \(18\) |
default | \(\frac {2 \left (x^{3}+1\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{9}\) | \(23\) |
risch | \(\frac {2 \left (x^{3}+1\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{9}\) | \(23\) |
elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x^{3} \sqrt {x^{3}+1}}{9}+\frac {2 \sqrt {x^{3}+1}}{9}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(53\) |
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none
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} \]
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\[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int x^{2} \sqrt {x + 1} \sqrt {x^{2} - x + 1}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{315} \, {\left ({\left (5 \, {\left (7 \, x - 23\right )} {\left (x + 1\right )} + 258\right )} {\left (x + 1\right )} - 213\right )} \sqrt {{\left (x + 1\right )}^{2} - 3 \, x} \sqrt {x + 1} + \frac {2}{105} \, {\left (3 \, {\left (5 \, x - 12\right )} {\left (x + 1\right )} + 71\right )} \sqrt {{\left (x + 1\right )}^{2} - 3 \, x} \sqrt {x + 1} \]
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Time = 11.90 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x^2 \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2\,\left (x^3+1\right )\,\sqrt {x+1}\,\sqrt {x^2-x+1}}{9} \]
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